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Unformatted text preview: Lecture 1 Pre-Derivatives Basics Stocks and bonds are referred to as underlying basic assets in financial markets. Nowadays, more and more derivatives are constructed and traded whose payoffs depend on the values of these underlying assets or other derivatives. We discuss some issues in this lecture before introducing derivatives. 1.1 Interest rates 1.1.1 Time value of money $100 today will be worth more than $100 next year. It simply means that you can increase the value of $100 by investing it. Albert Einstein once described compound interest as the greatest mathematical discovery of all time. The importance of interest rate of a riskless money market lies in its role as a yardstick for measuring the profitability of many risky investment strategies. Suppose stock A and stock B have average annual returns 4% and 10% respectively, while a CD (certificate of deposit) account in your local bank offers 5% annual rate. (You are obligated to keep the account for 5 years, say.) Which one would you choose: A, or B, or CD? It need not be easy to decide due to many factors involved. Still, here is a simplified answer. Using statistical terms, both mean and variance need to be taken into consideration. On the one hand, the CD is better than stock A and worse than stock B in terms of mean profit. On the other hand, greater attention should be paid to variance (usually referred to as volatility in financial jargon). Here the CD is regarded as a bench mark for comparison since it is risk-free, i.e. $100 deposited in the CD will for sure increase to $105 after one year. The actual profitability margin in stock B vs CD could be much greater than the mean difference 10% - 5% = 5%, or it is also possible to lose a lot of money in stock B. Both outcomes have positive probabilities to occur. It is this uncertainty that makes investment decisions more challenging. Later we will discuss in more detail the trade-off of return vs risk. For one thing, if we want to compare the payoffs of stock B and the CD seriously, we at least need to know the chances for stock B price to go up and down (by a certain amount) in a given time period. Let r a and r m denote the annual and monthly interest rates. It is a useful exercise to convert r a to an equivalent r m in the sense that any initial value P will end up with the same amount, following two different compounding rules (annually and monthly), i.e. P (1 + r a ) = P (1 + r m ) 12 . (1.1) Canceling P on both sides leads to 1 + r a = (1 + r m ) 12 , 1 hence r m = exp log(1 + r a ) 12 - 1 . (1.2) An approximation rule r m = r a / 12 (1.3) is often used. For r a = 6%, (1.2) and (1.3) will yield r m = 0 . 48676% and r m = 0 . 5% respectively....
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- Spring '08